Operators preserving the strong Birkhoff–James orthogonality on B(H)B(H)

We characterize the class of surjective (conjugate) linear mappings Φ:B(H)→B(H)Φ:B(H)→B(H) that preserve the strong Birkhoff–James orthogonality in both directions. We also give characterizations of rank-one operators as well as coisometries in terms of the strong Birkhoff–James orthogonality.